In standard persistence, the sublevel sets \(X_a=f^{-1}(-\infty ,a]\)  of the manifold \(X\)  are nested and PH is defined through the corresponding linear sequence of homology groups.
In extended persistence, we compute the homology of the sublevel sets, as well as the relative homology with respect to the superlevel sets \(X^a=f^{-1}[a,\infty )\) .
This is motivated by the fact that the relative homology group \(H_k(X,X^a)\)  of dimension \(k\)  is isomorphic to the cohomology group of \(X_a\)  of dimension \(dim-k\) , denoted \(H^{dim-k}(X_a)\) , where \(dim\)  is the dimension of the manifold \(X_a\)  [1]}.
